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Reverse Derivative Categories Cockett, Robin Cruttwell, Geoffrey Gallagher, Jonathan Lemay, Jean-Simon Pacaud MacAdam, Benjamin Plotkin, Gordon Pronk, Dorette Reverse Derivatives Cartesian Reverse Differential Categories Categorical Semantics Cartesian Differential Categories Dagger Categories Automatic Differentiation The reverse derivative is a fundamental operation in machine learning and automatic differentiation [Martín Abadi et al., 2015; Griewank, 2012]. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by [Blute et al., 2009] for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Robin Cockett and Geoffrey Cruttwell and Jonathan Gallagher and Jean-Simon Pacaud Lemay and Benjamin MacAdam and Gordon Plotkin and Dorette Pronk 2020 Is Part Of LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CSL.2020.18 urn:nbn:de:0030-drops-116611 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.18 eng https://creativecommons.org/licenses/by/3.0/legalcode